Introduction to coding theory (CMU, Spr 2010)

April 7, 2010

Lecture 20 summary

Filed under: Lecture summary — Venkat Guruswami @ 6:46 pm

We touched upon some aspects relating to the combinatorics of list decoding Reed-Solomon codes. We discussed the geometric intuition underlying Reed-Solomon decoding, and presented Sudan’s algorithm for polynomial reconstruction/list decoding RS codes. We analyzed it by optimizing individual degrees leading to an algorithm working for agreement parameter t > 2 \sqrt{kn} where k is the degree and n is the number of points. By optimizing the (1,k)-weighted degree, we improved this to t \ge \sqrt{2kn}.

We motivated the method of using multiple zeroes in the interpolation with an example, and then presented the details of how to enforce multiple (say w) zeroes at a point for a bivariate polynomial, why it buys us w zeroes for each point of agreement, and why this should improve the agreement t needed by a further \sqrt{2} factor thus matching the Johnson bound.

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